Real Analysis: Cardinality and Countability

Show that all even numbers are countable by mapping them onto the set of natural numbers.

Show that the set {$\frac{1}{1^2}$, $\frac{1}{2^2}$, $\frac{1}{3^2}$, ...} is countable by creating a bijection to the natural numbers.

Prove that the set of all real numbers between 0 and 1 is uncountable.

Determine whether a given set is countable by demonstrating a one-to-one correspondence with the positive integers.

Prove that the set of all rational numbers between 0 and 1 is countable.

Identify examples of uncountable sets within continuous subsets of the real line.

Is the set of even integers countable?

Given an output value in a one-to-one correspondence between natural numbers and even integers, determine the input value.

Prove that the cardinality of the natural numbers, denoted as $\aleph_0$, is equal to the cardinality of the set of all positive odd integers.

Prove that the set of rational numbers $\mathbb{Q}$ is countable by using the classical method of defining the height of a rational number and showing it as a countable union of finite sets.

Using Campbell's method with base 11, prove that the set of positive rational numbers $\mathbb{Q}^+$ is countable.

Establish a bijective correspondence between the integers and the rational numbers.

Prove that the set of real numbers between 0 and 1, where their decimal expansion only contains the digits 3 and 4, is uncountable using Cantor's diagonalization argument.

Prove Cantor's Theorem which states that for any set $A$, the power set of $A$ always has a cardinality strictly greater than $A$.